Question

In each exercise, use the stated information to determine the unspecified coefficients in the given differential equation. \(t=0\) is a regular singular point. The roots of the indicial equation at \(t=0\) are \(\lambda_{1}=1\) and \(\lambda_{2}=2\). \(t=0\) is a regular singular point. The roots of the indicial equation at \(t=0\) are \(\lambda_{1}=1+2 i\) and \(\lambda_{2}=1-2 i\). \(t=0\) is a regular singular point. One root of the indicial equation at \(t=0\) is \(\lambda=2\). The recurrence relation for the series solution corresponding to this root is \(\left(n^{2}+n\right) a_{n}-4 a_{n-1}=0, n=1,2, \ldots .\) \(n^{2} a_{n}-(n-1) a_{n-1}+3 a_{n-2}=0, n=2,3, \ldots .\) $$ { }^{2} y^{\prime \prime}+t(\alpha+2 t) y^{\prime}+\left(\beta+t^{2}\right) y=0 $$

Short answer

Based on the given information and using the Frobenius method, we are able to express the series solutions for each set of roots. However, there is not enough information to uniquely determine the values of the unspecified coefficients α and β in the last provided differential equation.

Step-by-step solution

The first given information has the roots \(\lambda_1 = 1\) and \(\lambda_2 = 2\). The corresponding differential equation is provided as: $$t^2y'' + P(t)y' + Q(t)y = 0$$ #2. Step 2: Use the roots to express the series solutions to the equation#

Using the roots \(\lambda_1\) and \(\lambda_2\), we can express the series solutions as: $$y(t) = \sum_{n=0}^{\infty} a_n t^{n+\lambda_1} + \sum_{n=0}^{\infty} b_n t^{n+\lambda_2}$$ #3. Step 3: Find recurrence relations between coefficients if possible#

In this case, there are no recurrence relations given for the coefficients from this set of roots. So we proceed to the next set of roots. #4. Step 1: Identify the given roots of the indicial equation and the corresponding differential equation#

The second given information has the roots \(\lambda_1 = 1+2i\) and \(\lambda_2 = 1-2i\). The corresponding differential equation is the same as before: $$t^2y'' + P(t)y' + Q(t)y = 0$$ #5. Step 2: Use the roots to express the series solutions of the given differential equation#

Using these complex roots, we can express the series solutions as follows: $$y(t) = \sum_{n=0}^{\infty} a_n t^{n+\lambda_1} + \sum_{n=0}^{\infty} b_n t^{n+\lambda_2}$$ #6. Step 3: Find recurrence relations between coefficients if possible#

In this case, there are no recurrence relations given for the coefficients from this set of roots. So we proceed to the next set of roots. #7. Step 1: Identify the given roots of the indicial equation and the corresponding differential equation#

The third given information has only one root \(\lambda = 2\). The corresponding differential equation is again in the same form: $$t^2y'' + P(t)y' + Q(t)y = 0$$ #8. Step 2: Use the roots to express the series solutions of the given differential equation#

Using the given root, we can express the series solution as: $$y(t) = \sum_{n=0}^{\infty} a_n t^{n+\lambda}$$ #9. Step 3: Find recurrence relations between coefficients if possible#

The given recurrence relation for this root is: $$\left(n^{2}+n\right) a_{n}-4 a_{n-1}=0, n=1,2, \ldots $$ #10. Step 4: Determine the unspecified coefficients in the given differential equation#

This final step is the same for all given information. However, for the given differential equation, we must find the values of \(\alpha\) and \(\beta\): $${ }^{2} y^{\prime \prime}+t(\alpha+2 t) y^{\prime}+\left(\beta+t^{2}\right)y=0$$ Unfortunately, this problem is not well-posed as it does not provide enough information to uniquely determine the values of the coefficients \(\alpha\) and \(\beta\) in this differential equation.

Key concepts

Indicial Equation

An indicial equation is a crucial concept in solving linear differential equations with singular points. When we encounter a point, such as \( t=0 \), which is a regular singular point, the indicial equation helps determine the behavior of solutions near this point. Essentially, it can be seen as a polynomial equation whose roots are derived from the exponents of the series solutions at the singular point.

Here's a simple breakdown of how indicial equations work:

• They are derived from the differential equation by substituting series solutions.
• The roots of the indicial equation tell us about the order of the dominant term in the solution's power series expansion.
• The roots can be real, complex, or multiple, which affects the nature of the series solution.

In the given exercise, the roots of the indicial equation at \( t=0 \) were provided as \( \lambda_1 = 1 \) and \( \lambda_2 = 2 \), as well as a set of complex roots \( \lambda_1 = 1+2i \), \( \lambda_2 = 1-2i \), indicating that both real and complex series solutions can exist for these indicial equations.

Series Solutions

Series solutions are a method of solving linear differential equations, particularly useful at singular points like \( t=0 \). This approach involves assuming a series format solution for the dependent variable in terms of the independent variable, which leads to expressing the solution as a power series:

1. A typical series solution looks like \( y(t) = \sum_{n=0}^{\infty} a_n t^{n+\lambda} \), where \( \lambda \) is a root of the indicial equation.
2. We use the series solution to find relations among the coefficients, \( a_n \), which will define the true solution.
3. In some cases, we express more complex series solutions due to complex roots such as \( y(t) = \sum_{n=0}^{\infty} a_n t^{n+\lambda_1} + \sum_{n=0}^{\infty} b_n t^{n+\lambda_2} \).

In the context of the exercise, the series solutions account for each set of roots of the indicial equation, ensuring all potential solutions, from simple real to the more involved complex scenarios, are considered.

Recurrence Relations

When solving differential equations using series solutions, recurrence relations serve as the connection pathways between the coefficients \( a_n \) in the series.

These relations help us determine each subsequent coefficient based on the previous ones, usually emerging from substituting the series into the original differential equation and equating terms:

• A recurrence relation defines \( a_{n} \) in terms of \( a_{n-1} \), and sometimes, earlier coefficients like \( a_{n-2} \).
• Such relations are often linear but may involve multiplicative constants or summations that need careful computation.
• In the provided exercise, for a root \( \lambda = 2 \), the recurrence relation is \((n^2 + n)a_n - 4a_{n-1} = 0\), guiding us in computing specific coefficients for the series solution.

Understanding and solving these relations allow us to build the full series, providing an exact solution to the differential equation when summed appropriately.